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Model Theoretic Perspectives on the Philosophy of Mathematics John T. Baldwin∗ Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago February 10, 2010 And some infinites are larger than other infinites and some are smaller. Robert Grosseteste 13th century [Fre54] We approach the `practice based philosophy of logic' by examining the practice in one specific area of logic, model theory, over the last century. From this we try to draw lessons not for the philosophy of logic but for the philosophy of mathematics. We argue in fact that the philosophical impact of the developments in mathematical logic during the last half of the twentieth century were obscured by their mathematical depth and by the intertwining with mathematics. That is, that concepts which are normally regarded by both mathematicians and philosophers as `simply mathematics' have philosophical importance. We make two claims. First is that the mere fact that logical methods have had mathematical impact is important for any investigation of mathematical methodology. Twentieth century logic introduced techniques that were important not just for the problems they were originally designed to solve (arising out of Hilbert's program) but across broad areas of mathematics. But, from a philosophical standpoint, there is a further impact. These methods actually provide tools for the analysis of mathematical methodology. We view the practice-based philosophy of `Subject X' as a broad inquiry into and critical analysis of the conceptual foundations of actual work in subject X1. The topic of this workshop was the (practice-based) philosophy of logic and mathematics. These are clearly different areas2; there is a substantial overlap in mathematical logic. By mathematical logic we mean the use of formalization of mathematical arguments and concepts to investigate both mathematics and philosophical questions about mathematics. Emphasizing 20th century model theory, we describe below some aspects of the development of mathematical logic since its foundation by such as Boole, Frege, Pierce, and Shr¨oderin the late 19th century. Thus we restrict to the study of formalization of declarative sentences about mathematical objects3. Other areas of logic, which study more complex natural language sentences (tense, deontic, modal, etc.) and often use mathematical tools, are not discussed here. The longtime standard definition of logic is `the analysis of methods of reasoning'. This does not describe the perspective of a contemporary model theorist. A model theorist is a self-conscious mathematician. A model theorist uses various formal languages and semantics to prove mathematical theorems. But there is an inherently ∗We give special thanks to the Mittag-Leffler Institute where we were able to rethink and focus the ideas of this talk. Baldwin was partially supported by NSF-0500841. The paper builds on a presentation at Notre Dame in Fall of 2008. 1This is a paraphrase of part of Dutilh-Novaes presentation. 2Even if some form of the logicism program had succeeded, as we argue below the logical foundations of mathematics do not exhaust the philosophy of mathematics. 3The ontological status of these objects is not relevant to our analysis. 1 metamathematical aspect. The very notion of model theory involves seeking common patterns across distinct areas of mathematical investigation. One of our goals below is to make this notion of `distinct area' more precise. In particular, we view the practice-based philosophy of mathematics as a broad inquiry into and critical analysis of the conceptual foundations of actual mathematical work. This investigation also includes a study of the basic methodologies and proof techniques of the subject4. The foundationalist goal of justifying mathematics is a part of this study. But the study we envision cannot be carried out by interpreting the theory into an ¨uber theory such as ZFC; too much information is lost. The coding does not reflect the ethos of the particular subject area of mathematics. The intuition behind fundamental ideas such as homomorphism or manifold disappears when looking at a complicated definition of the notion in a language whose only symbol is . Tools must be developed for the analysis and comparison of distinct areas of mathematics in a way that maintains meaning; a simple truth preserving transformation into statements of set theory is inherently inadequate. The traditional foundationalist approach sacrifices explanation on the altar of justification. The discussion below has both a sociological and a philosophical aspect. Sociologically, in this introduction, we describe recent examples illustrating model theoretic practice. Philosophically, in the two main sections of the paper we propose some tools for studying the methodology of mathematics. We outline a program for using model theoretic concepts for a) formalizing a notion of `area of mathematics' and b) analyzing basic concepts of mathematics. In Section 1) we sketch the history of model theory in the twentieth century and in particular the development of the notion of a complete theory. We argue for the notion of a first order complete theory as a useful unit of analysis for describing `an area of mathematics'. We conclude the historical discussion with an introduction to the sophisticated model theoretic methods developed in the last 40 years. In Section 2) we discuss how these methods can provide insight into the way fundamental notions are specified in different areas of mathematics. We glimpse the use of model theoretic notions to analyze one particular mathematical notion: dimension. In both cases a) and b) , our main point is that model theoretic tools can be brought to bear. We are simply giving introductory sketches of illustrations of that thesis. In the sociological mode, we now describe some papers at the Mid-Atlantic Model Theory conference held in the Fall of 2008 at Rutgers. In the late 70's a large gap was seen between EC `East Coast' and WC `West Coast' model theory: a) EC used model theory in various parts of mathematics and b) WC developed an independent subject area of `model theory'. We elaborate this contrast below. But the current situation is best summed up by Pillay's affirmation at the 2000 ASL panel on the future of logic { `there is only one model theory.'5 At least seven of the ten papers at the Rutgers conference, even those focused on algebraic notions such as non- archimedean geometry, semi-abelian varieties, or difference fields integrate the deep concepts developed in the pure theory since 1970. For example, one paper concerned a conjecture of Cherlin which uses model theoretic concepts to lift the program of the classification of finite simple groups to the classification of simple groups of finite Morley rank. Even the statement of the problem is posed in model theoretic terms that provide a way to organize topics that were already in the mathematical air. The investigation involves significant techniques from model theory, finite group theory, and algebraic groups. Even the relatively few papers at this conference that were `pure' developed concepts central to current research in e.g. the model theory of valued fields6. Our summary of this conference has focused on `main-stream' contemporary model theory7 which develops logical 4For a broad investigation in the philosophy of mathematics including a study of leading contemporary mathematicians (e.g. Grothendieck, Langlands, Shelah, Zilber) see [Zal09] 5This is my recollection of Pillay's oral statement. He describes the ‘unification’ at considerable length in his contribution to [BKPS01]. 6Shelah's concept of theories without the independence property (nip or dependent depending on the author) were expounded in the least-applied talk. Hrushovski's paper `Stable groups and approximate group theory'[Hru09], which uses the model theoretic analysis of these theories as a tool for the study of groups, was the subject of semester-long seminars at UCLA, Berkeley, Urbana, and Leeds in the Fall of 2009. Fields medalist Terrence Tao discusses the progress of the UCLA seminar in the blog at http: //terrytao.wordpress.com/. 7It does not encompass a number of other areas of model theory such as models of arithmetic, finite model theory, model theory in computer science, higher order and other extensions of first order logic, and universal algebra. 2 techniques of model theory and integrates them in the investigation of problems across mathematics. 1 Historical Survey of model theory The integral connections of model theory with modern mathematics as described in the introduction are often (and often correctly) seen as a falling away from philosophical concerns. But as we'll see below, many of these interactions do stem from concerns about explanation and coherence of mathematical ideas that have a philosophical basis. The divorce is from narrow concern with the formal justification of results. Moreover, natural philosophical issues arise from some technical results. For example, there are vast differences between the role of @0 and any uncountable cardinal in the study of categoricity. What is so different about countability? After a survey of the history of model theory I expound the use of model theoretic concepts as a tool for such an analysis of the foundations of mathematics. We review this history from a standpoint similar to this paper but with an emphasis on the mathematical applications in [Bal10]. For lack of space, I concentrate on first order logic; recent work in infinitary logic [Bal09, She09, Zil04] contributes new problems and methods to our analysis. We distinguish three types of analysis in first order model theory: 1. Properties of first order logic (1930-1965) 2. Properties of complete theories (1950-present) 3. Properties of classes of theories (1970-present) 1.1 Properties of first order logic (1930-1965) The essence of model theory is a clear distinction between syntax and semantics. Sentences in a formal lan- guage for a vocabulary τ are true or false in structures for τ.

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